Bechir Dali

Construction of canonical coordinates for exponential Lie groups

Given an exponential Lie group G, we show that the constructions of B. Currey, 1992, go through for a less restrictive choice of the Jordan-Holder basis. Thus we obtain a stratification of g * into G-invariant algebraic subsets, and for each such subset Ω, an explicit cross-section Σ C Ω for coadjoint orbits in Ω, so that each pair (Ω, Σ) behaves predictably under the associated restriction maps on g * . The cross-section mapping σ: Ω → Σ is explicitly shown to be real analytic. The associated Vergne polarizations are not necessarily real even in the nilpotent case, and vary rationally with ∈ Ω. For each Ω, algebras e 0 (Ω) and e 1 (Ω) of polarized and quantizable functions, respectively, are defined in a natural and intrinsic way. Now let 2d > 0 be the dimension of coadjoint orbits in Ω. An explicit algorithm is given for the construction of complex-valued real analytic functions {q 1 , q 2 , ..., q d } and {p 1 ,p 2 ,...,p d } such that on each coadjoint orbit O in Ω, the canonical 2-form is given by Σdp k ^ dq k . The functions {q 1 ,q 2 , ..., q d } belong to e 0 (Ω), and the functions {p 1 ,p 2 , ... ,p d } belong to e 1 (Ω). The associated geometric polarization on each orbit O coincides with the complex Vergne polarization, and a global Darboux chart on O is obtained in a simple way from the coordinate functions (p 1 , ... , p d , q 1 , ... , q d ) (restricted to O). Finally, the linear evaluation functions l ↦ l(X) are shown to be quantizable as well.

The spaces Hn(osp(1∣2),M) for some modules M

We first generalize a result by Bavula on the sl(2) cohomology to the osp(1∣2) cohomology and then we entirely compute the cohomology for a natural class of osp(1∣2) modules M. We study the restriction to the sl(2) cohomology of M and apply our results to the module M=Dλ,μ of differential operators on the superline acting on densities.

Deformation of vect(1)-modules of symbols

We consider the action of the Lie algebra of polynomial vector fields,

Local convergence of Newton's method on the Heisenberg group

\mathfrak{vect}(1)

Deformation of -modules of symbols

, by the Lie derivative on the space of symbols

Deformation of vect(1)vect(1)-modules of symbols

\mathcal{S}_\delta^n=\bigoplus_{j=0}^n \mathcal{F}_{\delta-j}

Deformation of Vect(

. We study deformations of this action. We exhibit explicit expressions of some 2-cocycles generating the second cohomology space

\mathbb{R})

\mathrm{H}^2_{\rm diff}(\mathfrak{vect}(1),{\cal D}_{\nu,\mu})

-Modules of Symbols

where

Canonical coordinates for a class of solvable groups

{\cal D}_{\nu,\mu}